Why Don’t Ions Neutralize in an Applied Current Solution?

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When a current is applied to a solution containing mobile ions, the expectation might be that cations would lose electrons and anions would gain electrons, resulting in the formation of neutral atoms, similar to electrolysis. However, the ions remain charged due to several factors. One key reason is that the potential applied may not be sufficient to drive the necessary redox reactions for neutralization. Additionally, the presence of a stable ionic environment in the solution maintains the ions' charge. The energy barrier for the redox reactions may be too high, preventing the ions from gaining or losing electrons effectively. This results in the formation of a charged layer rather than neutral atoms.
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If a current is being applied to a solution with mobile ions in it, why don't the cations lose electrons and the anions gain electrons creating neutral atoms like they do in electrolysis, what causes these ions to stay charged and form a charged lager when they could just gain/lose charge and become neutral?
 
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The potential not being high enough for the redox reaction to happen?
 
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